LLL and other lattice reduction techniques (such as PSLQ) try to find a short basis vector relative to the 2-norm, i.e. for a given basis that has $ \varepsilon $ as its shortest vector, $ \varepsilon \in \mathbb{Z}^n $, find a short vector s.t. $ b \in \mathbb{Z}^n, ||b||_2 < ||c^n \varepsilon||_2 $.

Has there been any work done to find short vectors based on other, potentially higher, norms? Is this a meaningful question?

There is an LLL analogue for arbitrary norms; the original paper by Lovász and Scarf can be found here. I recently found a bachelor thesis on lattice reduction in infinity norm, which contains several other references (for example, work by Kaib and Ritter).